16edo: Difference between revisions

| de = 16-EDO

16edo's step size is sometimes called an '''eka''', a term proposed by [[Luca Attanasio]], from Sanskrit [[wikt:%E0%A4%8F%E0%A4%95#Sanskrit|एक]] (''éka'', "one", "unit"),<ref>[http://www.armodue.com/risorse.htm Armodue: le risorse di un nuovo sistema musicale]</ref> when used as an [[interval size unit]], especially in the context of [[Armodue]] theory.

The [[3/2|perfect fifth]] of 16edo is 27 cents flat of 3/2, flatter than that of [[7edo]] so that it generates an [[2L 5s|antidiatonic]] instead of [[5L 2s|diatonic]] scale, but sharper than [[9edo]]'s fifth, to which it similarly retains the characteristic of being a fifth while being distinctly flat of 3/2. If the fifth is interpreted as 3/2, this befits a tuning of [[mavila]], the [[5-limit]] [[regular temperament|temperament]] that [[tempering out|tempers out]] [[135/128]], such that a stack of four fifths gives a [[6/5]] minor third instead of the familiar [[5/4]] major third as in [[meantone]]. A more accurate restriction is [[mabilic]], which discards the inaccurate mapping of 3 while keeping the fifth as a generator.

This leads to some confusion in regards to interval names, as what would be major in diatonic now sounds minor; there are several ways to handle this (see in [[#Intervals]]).  

In general, 16edo tends to better approximate the differences between odd [[harmonic]]s than odd harmonics themselves, though it has a [[5/1|5th harmonic]] which is only 11 cents flat, and a [[7/1|7th harmonic]] which is only 6 cents sharp. As such, 16edo can be seen as an approach to tuning that takes advantage of the idea that simpler ratios can be functionally approximated with greater error (i.e. a 3/2 that's 25 cents flat is still recognizable, but a 5/4 that's 25 cents flat loses much of its identity and a 7/4 that's 25 cents flat is completely unrecognizable). In essence, 16edo's 3, 5, and 7 are backwards from 12edo's, with 7 being nearly perfect, 5 being decent, and 3 being distinctly out-of-tune.  

In terms of higher primes, both 11 and 13 are approximated very flat, with the [[11/8]] not distinguished from [[4/3]], and [[13/8]] not distinguished from [[8/5]]. 16edo represents the no-9 no-15 [[25-odd-limit]] [[consistent]]ly, however.

Four steps of 16edo gives the 300{{c}} minor third interval shared by [[12edo]] (and other multiples of [[4edo]]), which approximates [[6/5]], and thus tempers out 648/625, the [[diminished comma]]. This means that the familiar [[diminished seventh chord]] may be built on any scale step with four unique tetrads up to [[octave equivalence]]. The minor third is of course not distinguished from the septimal subminor third, [[7/6]], so [[36/35]] and moreover [[50/49]] are tempered out, making 16edo a possible tuning for [[diminished (temperament)|septimal diminished]]. Another possible interpretation for this interval is the 19th harmonic, [[19/16]].

16edo shares several similarities with 15edo. They both share mappings of [[8/7]], [[5/4]], and [[3/2]] in terms of edosteps – in fact, they are both [[valentine]] tunings, and thus [[slendric]] tunings. 16edo and 15edo also both have three types of seconds and two types of thirds (not including arto/tendo thirds). However, 15edo's fifth is sharp while 16's is flat.  

16edo works as a tuning for [[extraclassical tonality]], due to its ultramajor third of 450 cents.

{{Harmonics in equal|16}}

Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and gamelans, with [[25edt]], [[41ed6]], and [[57ed12]] being good options.

=== Composition theory ===

* [[User:VectorGraphics/16edo theory|Vector's approach]]

== Intervals ==

Alternatively, one can use Armodue nine-nominal notation.

| 19/14, 27/20, 35/26, 256/189

|-

|-

|-

| super-5th,<br>subminor 6th

|-

|-

|-

|-

|-

| 27/14, 52/27

|}

<nowiki />* Based on treating 16edo as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.

16edo notation can be easy utilizing [[Goldsmith's Circle]] of keys, nominals, and respective notation{{clarify}}. The nominals for a 6 line staff can be switched for [[Erv Wilson]]'s Beta and Epsilon additions to A–G. The Armodue model uses a 4-line staff for 16edo.

Mos scales like Mavila[7] (or "inverse/anti-diatonic" which reverses step sizes of diatonic from LLsLLLs to ssLsssL in the heptatonic variation) can work as an alternative to the traditional diatonic scale, while maintaining conventional A–G ♯/♭ notation as described above. Alternatively, one can utilize the Mavila[9] mos, for a sort of "hyper-diatonic" scale of 7 large steps and 2 small steps. [[Armodue theory|Armodue notation]] of 16edo "Mavila[9] Staff" does just this, and places the arrangement (222122221) on nine white "natural" keys of the 16edo keyboard. If the 9-note (enneatonic) mos is adopted as a notational basis for 16edo, then we need an entirely different set of interval classes than any of the heptatonic classes described above; perhaps it even makes sense to refer to the octave ([[2/1]]) as the "[[decave]]". This is identical to the KISS notation for this scale when using numbers.

rect 471 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

[http://www.armodue.com/ricerche.htm Armodue]: Pierpaolo Beretta's website for his Armodue theory for 16edo (esadekaphonic), including compositions.

For resources on the Armodue theory, see the [[Armodue]] on this wiki

16edo chords can be named using ups and downs. Using diatonic interval names, chord names bear little relationship to the sound: a minor chord (spelled {{dash|A, C, E|med}}) sounds like [[4:5:6]], the classical major triad, and a major chord (spelled {{dash|C, E, G|med}}) sounds like [[10:12:15]], a classical minor triad! Instead, using antidiatonic names, the chord names will match the sound&mdash;but finding the name from the spelling follows the rules of antidiatonic rather than diatonic interval arithmetic.

| 4:5:6

| 10:12:15

| {{dash|0, 4, 9, 13|med}}

That is, reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected.

Examples can be found at the bottom of the page.

[[File:16ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:16ed2-001.svg|16ed2-001.svg]]

== Octave theory ==

16edo is also a tuning for the [[Jubilismic clan|no-threes 7-limit temperament tempering out 50/49]]. This has a period of a half-octave (600{{c}}), and a generator of a flat septimal major 2nd, for which 16edo uses 3\16. For this, there are mos scales of sizes 4, 6, and 10; extending this temperament to the full 7-limit can produce either Lemba or Astrology (16edo supports both, but is not a very accurate tuning of either).

16edo is also a tuning for the no-threes 7-limit temperament tempering out [http://x31eq.com/cgi-bin/uv.cgi?uvs=%5B-19%2C7%2C1%3E&limit=2_5_7 546875:524288], which has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "'''Magic family of scales'''".

 

 

"''16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh.''"

 

From a harmonic series perspective, if we take 13\16 as a 7/4 ratio approximation, sharp by 6.174{{c}}, and take the 300{{c}} minor third as an approximation of the harmonic 19th ([[19/16]], approximately 297.5{{c}}), that can combine with the approximation of the harmonic seventh to form a 16:19:28 triad .

 

The interval between the 28th &amp; 19th harmonics, 28:19, measures approximately 671.3{{c}}, which is 3.7{{c}} away from 16edo's "narrow fifth". Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7{{c}} just, 525.0{{c}} in 16edo). A perhaps more consonant open voicing is 7:16:19

 

 

=== Commas ===

 

{| class="commatable wikitable center-all left-3 right-4 left-6"

! [[Harmonic limit|Prime<br>limit]]

! [[Ratio]]<ref group=note>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

! [[Cent]]s

! Name

|-

| {{monzo| -7 3 1 }}

|-

| {{monzo| 3 4 -4 }}

|-

| {{monzo| -10 -1 5 }}

|-

| {{monzo| 23 6 -14 }}

| 3.34

| [[36/35]]

| {{monzo| 2 2 -1 -1 }}

|-

| [[525/512]]

| {{monzo| -9 1 2 1 }}

|-

| {{monzo| 1 0 2 -2 }}

|-

| 7

| [[64827/64000]]

| {{monzo| -9 3 -3 4 }}

| Laquadzo-atrigu

|-

| {{monzo| 0 -2 5 -3 }}

| 21.18

| Triru-aquinyo

|-

| {{monzo| 1 2 -3 1 }}

|-

| {{monzo| -10 1 0 3 }}

|-

| {{monzo| 11 1 -3 -2 }}

|-

| [[121/120]]

| {{monzo| -3 -1 -1 0 2 }}

|-

| [[176/175]]

| 9.86

|-

| {{monzo| -7 -1 1 1 1 }}

|-

| {{monzo| -3 2 -1 2 -1 }}

| [[3025/3024]]

| {{monzo| -4 -3 2 -1 2 }}

| 0.57

 

=== Rank-2 temperaments ===

* [[List of 16et rank two temperaments by badness]]

 

{| class="wikitable center-1 center-2"

|-

| 1\16

| [[Magic]]/[[muggles]]

| 7\16

| [[Mavila]]/[[armodue]]

|-

|-

|-

| [[Diminished (temperament)|Diminished]]/[[demolished]]

| 1\16

 

* [[magic]] anti-diatonic 3L4s 1414141 (5\16, 1\1)

* [[magic]] superdiatonic 3L7s 1311311311 (5\16, 1\1)

* [[magic]] chromatic 11121121112 3L10s (5\16, 1\1)

* [[mavila]] anti-diatonic 2L5s 2223223 (9\16, 1\1)

 

 

 

{| class="wikitable"

| [5]:

| [7]:

|[[File:MavilaAntidiatonic16edo.mp3]]

| [9]:

|[[File:MavilaSuperdiatonic16edo.mp3]]

 

 

{| class="wikitable"

|-

| [8]:

|-

| [12]:

 

 

[7]: 1 4 1 4 1 4 1

 

[10]: 1 3 1 1 3 1 1 1 3 1

 

[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1

 

'''Cynder/Gorgo'''

 

[5]: 3 3 4 3 3

 

[6]: 3 3 1 3 3 3

 

[11]: 1 2 1 2 1 2 1 2 1 2 1

 

 

 

 

 

== Metallic harmony ==

In 16edo, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16edo approximates 7/4 well enough to use

 

it in place of the usual 3/2, and in Mavila[7] this 7/4 approximation shares an interval class with a well-approximated 11/6 (at 1050{{c}}). Stacking these two intervals reaches 2025{{c}}, or a minor 6th plus an octave. Thus the out-of-tune 675{{c}} interval is bypassed, and all the dyads in the triad are consonant.

 

Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, {{nowrap|{{dash|0, 975, 2025{{c}}}}}}, and a large one, {{nowrap|{{dash|0, 1050, 2025{{c}}}}}}. William Lynch, a major proponent of this style of harmony, calls these two triads "hard" and "soft", respectively. In addition, two other "symmetrical" triads are also obvious possible chords: a narrow symmetrical triad at {{nowrap|{{dash|0, 975, 1950{{c}}}}}}, and a wide symmetrical triad at {{nowrap|{{dash|0, 1050, 2100{{c}}}}}}. These are sort of analogous to "diminished" and "augmented" triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them "Metallic triads".

 

The ssLsssL mode of Mavila[7] contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords are wide symmetrical triads 0-1050-2025{{c}}. In Mavila[9], hard and soft triads cease to share a triad class, as 975{{c}} is a major 8th, while 1050{{c}} is a minor 9th; the triads may still be used, but parallel harmonic motion will function differently.

 

Another possible MOS scales for this approach would be Lemba[6], which gives two each of the soft, hard, and narrow symmetric triads.

 

 

'''16-tone piano layout based on the mavila[7]/antidiatonic scale'''

 

This Layout places mavila[7] on the black keys and mavila[9] on the white keys, according to antidiatonic notation.

 

[[File:16-EDO-PIano-Diagram.png|alt=16-EDO-PIano-Diagram.png|748x293px|16-EDO-PIano-Diagram.png]]

 

 

A visualization of 16edo being two interleaved copies of [[8edo]] and four interleaved copies of [[4edo]].

 

[[File:16edo_wheel_01.png|alt=16edo wheel 01.png|325x325px|16edo wheel 01.png]]

 

 

 

! colspan="2" |Diatonic (i.e. 12edo)

! colspan="2" |Antidiatonic (i.e. 16edo)

|-

|-

|-

| A major chord

| what chord is D F A♯?

| what chord is C E G♭ B♭?

| C + M3 + A5 + A7

| C major scale = C + M2 + M3<br>+ P4 + P5 + M6 + M7 + P8

| C D♯ E♯ F<br>G A♯ B♯ C

| C D♭ E♭ F<br>G A♭ B♭ C

| C minor scale = C + M2 + m3<br>+ P4 + P5 + m6 + m7 + P8

| C D♯ E F<br>G A B C

| C + m2 + M3 + P4<br>+ P5 + M6 + M7 + P8

| C D♭ E F<br>G A B C

| what scale is A B♯ C♭ D<br>E F G♭ A?

| A + M2 + m3 + P4<br>+ P5 + M6 + m7 = A dorian

| A B♭ C♯ D<br>E F G♯ A

| A + m2 + M3 + P4<br>+ P5 + m6 + M7

 

 

; [[Abnormality]]

* [https://www.youtube.com/watch?v=zao6E8GdQh0 ''it's not not opposite day''] (2023)

* [https://www.youtube.com/watch?v=1pa3dztk8o0 ''nightfall''] (2024)

 

; [[Beheld]]

* [https://www.youtube.com/watch?v=kzPeVB2mncc ''Nebulous vibe'']

 

* [https://cityoftheasleep.bandcamp.com/track/huckleberry-regional-preserve ''Huckleberry Regional Preserve'']

* [https://cityoftheasleep.bandcamp.com/track/illegible-red-ink ''Illegible Red Ink'']

 

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/IfVvjoRqqNk ''16edo jam''] (2025)

* [https://www.youtube.com/watch?v=cUgbkkIvy0g ''Waltz in 16edo''] (2025)

 

; [[E8 Heterotic]]

* [https://youtu.be/a8Jgb_XIj7c "Hexed"]

 

; [[Fabrizio Fiale]]

* [https://www.soundclick.com/music/songInfo.cfm?songID=12370649 ''Prenestyna Highway'']

* [https://www.soundclick.com/music/songInfo.cfm?songID=7715803 ''Palestrina Morta, fantasia quasi una sonata'']

* [https://soundcloud.com/fff-fiale/in-sospensione-neutra ''In Sospensione Neutra'']

 

; [[Aaron Andrew Hunt]]

* [https://soundcloud.com/uz1kt3k/fuga-a3-in-16et ''Fuga a3 in 16ET'']

 

; [[Last Sacrament]]

* [http://lastsacrament.bandcamp.com/album/enantiodromia ''Enantiodromia''] (album) (from 2013)

* [https://lastsacrament.bandcamp.com/album/maniacal-meditations-ep ''Maniacal Meditations''] (EP) (2013 EP)

 

; [[William Lynch]]

* [[:File:Mavila_Jazz_Rhodes_1.mp3|''Mavila Jazz Groove'']]

* [[:File:mavila4.mp3|''Cold, Dark Night for a Dance'']]

 

* [https://www.youtube.com/watch?v=vIWxP_C0aUM ''Mavila Fugue'']

* [https://www.youtube.com/watch?v=KYkmT46oGhw ''Canon at the Semitone on The Mother's Malison Theme'', for Cor Anglais and Violin] ([https://www.youtube.com/watch?v=I6BUauD8EaE for Organ])

* [https://www.youtube.com/watch?v=P7LUSRd1kMg ''Canon on Twinkle Twinkle Little Star'', for Organ] (2023) ([https://www.youtube.com/watch?v=QHJYyqge_JQ for Baroque Oboe and Viola])

* [https://www.youtube.com/shorts/I4-URAGgQMQ ''Baroque Micropiece in 16edo''] (2024)

 

; [[Herman Miller]]

* [http://www.io.com/%7Ehmiller/midi/16tet.mid ''Etude in 16-tone equal tuning'']{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3 play]{{dead link}} ([http://soonlabel.com/xenharmonic/archives/2604 organ version]{{dead link}})

 

; [[Nae Ayy]]

* [https://www.youtube.com/watch?v=H74psBvdeT4 ''Rambling'']

* [https://www.youtube.com/watch?v=OAhV8ol2Hbw ''a n g e r y'']

* [https://www.youtube.com/watch?v=-MboZelse90 ''Maundering'']

 

; [[NullPointerException Music]]

* [https://www.youtube.com/watch?v=LXsZIbT6wpM ''Edolian - Seventhic''] (2020)

* [https://www.youtube.com/watch?v=UrQPr7V9feA ''Finality''] (2021)

 

; [[Jean-Pierre Poulin]]

* [http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3 ''Armodue78'']

 

; [[Ron Sword]]

* [https://soundcloud.com/ron-sword/mavila-fog ''The Foggy Road from Pasadena'']{{dead link}}

 

; [[Chris Vaisvil]]

* [http://micro.soonlabel.com/16-ET/20120527-16-malathion.mp3 ''Malathion''] - [http://chrisvaisvil.com/?p=2358 details]

* [http://micro.soonlabel.com/16-ET/20130216_16edo_vesta.mp3 ''Being of Vesta''] - [http://chrisvaisvil.com/?p=3061 details]

* [http://micro.soonlabel.com/simultaneous-tunings/20130607_thin_ice_christiane.mp3 ''Thin Ice''] - [http://chrisvaisvil.com/?p=3354 details]

 

; [[Stephen Weigel]]

* [https://www.youtube.com/watch?v=0t7ZmlmrE0Q ''Shot Fades the Sum Of'']

* [https://www.youtube.com/watch?v=2y01AlgOPvk ''When the Saints go Marching'']

 

; [[Randy Winchester]]

* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3 Comets Over Flatland 5]{{dead link}}

 

; [[Woyten]]

* [https://www.youtube.com/watch?v=LLgClI8pyNw ''Don't Take Five''] (2021)

 

; [[Xotla]]

* "Robotic Dialogue" from ''Microtones & Garden Gnomes'' (2017) [https://xotla.bandcamp.com/track/robotic-dialogue-16edo Bandcamp] | [https://youtu.be/sFxny2JNGpo?si=8MKPuIMCR_Xx1DTi YouTube]

* "Cognitive Climate" from Science Fraction (2022) [https://open.spotify.com/track/52v382I0OUotQjHo0pPoXs Spotify] | [https://xotla.bandcamp.com/track/cognitive-climate-16edo Bandcamp] | [https://youtu.be/dNBDG4wymN8?si=XGbpNkRp3qUo0Xgb YouTube]

 

; [[User:Nick_Vuci|Nick Vuci]]

* [https://en.xen.wiki/images/4/44/NickVuci-20220206-16edo-Prelude.mp3 ''Prelude'']

* [https://en.xen.wiki/images/9/9a/NickVuci-20231102-16edo-SofterForJ.mp3 ''Softer for J'']

* [https://en.xen.wiki/images/4/48/NickVuci-20220306-16edo-Invention.mp3 ''2-Part Invention'']

* [https://en.xen.wiki/w/User:Nick_Vuci#Modal_Studies ''Mavila Modal Studies'']

* [https://en.xen.wiki/images/c/c6/NV-20210526-16NEJI128-SerialismDubstepSketch.mp3 ''EDM based on a tone row'']

 

; [[Zewen Senpai]]

* [https://www.youtube.com/watch?v=QOzBGd64Pi4 ''Simple Ambient Study No. 1'']

 

<references group=note/>

 

* [[57ed12]] - octave stretched version of 16edo; 57ed12 improves 3.5.11.13.17 but damages 2.7

 

=== Approaches ===

* [[Armodue theory]]

 

== References ==

<references />

 

== Further reading ==

* [[Sword, Ron]]. ''[https://ronsword.bigcartel.com/product/esadekaphonic-scales-for-guitar Hexadecaphonic Scales for Guitar: A Microtonal Guitar Method Book, for Theory, Scales, and Information on the Sixteen Equal Division Octave System]''. 2009. (semi-diminished fourth tuning)

* Sword, Ron. ''[http://www.metatonalmusic.com/books.html Hexadecaphonic Scales for Guitar: Theory, Scales and Information on the Sixteen Equal Division Octave system]''. 2010? (superfourth tuning)

* Sword, Ron. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011{{citation needed}}

 

[[Category:Teentuning]]

[[Category:Listen]]

[[Category:Pages with internal sound examples]]